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The idea is to integrate by parts twice. Start with the original integral,
∫sin(x)e^(2x) dx =
Let u = e^(2x) and dv = sin(x). Then du = 2e^(2x) and v = -cos(x). Apply integration by parts to get:
∫u dv = uv - ∫v du
∫e^(2x)sin(x) dx = -e^(2x)cos(x) - ∫-cos(x)2e^(2x) dx
∫e^(2x)sin(x) dx = -cos(x)e^(2x) + 2∫cos(x)e^(2x) dx
Call this last equation *1*. Now this seems to have made the integral we couldn't compute, ∫e^(2x)sin(x) dx, into yet another integral we can't compute, namely ∫cos(x)e^(2x) dx. So, apply integration by parts again - this time on just that rightmost integral.
For ∫cos(x)e^(2x) dx, let u = e^(2x) and dv = cos(x). Then du = 2e^(2x) and v = sin(x). So, from integration by parts you get:
∫u dv = uv - ∫v du
∫e^(2x)cos(x) dx = sin(x)e^(2x) - ∫sin(x)2e^(2x) dx
∫e^(2x)cos(x) dx = sin(x)e^(2x) - 2∫sin(x)e^(2x) dx
Now plug this substitution of the integral ∫e^(2x)cos(x) dx into equation *1* and see if it becomes cleaner.
∫e^(2x)sin(x) dx = -cos(x)e^(2x) + 2∫cos(x)e^(2x) dx
∫e^(2x)sin(x) dx = -cos(x)e^(2x) + 2[sin(x)e^(2x) - 2∫sin(x)e^(2x) dx]
∫e^(2x)sin(x) dx = -cos(x)e^(2x) + 2sin(x)e^(2x) - 4∫sin(x)e^(2x) dx
∫e^(2x)sin(x) dx + 4∫sin(x)e^(2x) dx = -cos(x)e^(2x) + 2sin(x)e^(2x)
5∫e^(2x)sin(x) dx = -cos(x)e^(2x) + 2sin(x)e^(2x)
5∫e^(2x)sin(x) dx = e^(2x)(2sin(x) - cos(x))
∫e^(2x)sin(x) dx = e^(2x)(2sin(x) - cos(x))/5
Hope this helps you!